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The research addresses the problem of maximizing a zero-one quadratic function. The report falls into three main sections.
The first section uses results from Hammer [12] and Picard and Ratliff [23] to develop a new test for fixing the value of a variable in some solution and to provide a means for calculating a new upper bound on the maximum of the function. In addition the convergence of the method of calculation for the bounds is explored in an investigation of its sharpness.
The second section proposes a branch and bound algorithm that uses the ideas of the first along with a heuristic solution procedure. It is shown that one advantage of this is that it may now be possible to identify how successful this algorithm will be in finding the maximum of a specified problem.
The third section gives a basis for a new heuristic solution procedure. The method defines a concept of gradient which enables a simple steepest ascent technique to be used. It is shown that in general this will find a local maximum of the function. A further procedure to help distinguish between local and global maxima is also given.

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